•
ASYMPTOTIC NORMALITY OF CONDITIONAL INTEGRALS
OF DIFFUSION PROCESSES
MONTSERRAT FUENTES
NCSU MIMEO SERIES #2520
NOVEMBER 3, 1999
ASYMPTOTIC NORMALITY OF CONDITIONAL INTEGRALS
OF DIFFUSION PROCESSES
MONTSERRAT FUENTES
I
North Carolina State University
Consider predicting the integral of a diffusion process Z in a bounded interval A,
based on the observations Z(tI n ), ... , Z(t nn ), where tIn, ... ,tnn is a dense triangular
array of points (the step of discretization tends to zero as n increases) in the bounded
interval. The best linear predictor is generally not asymptotically optimal. Instead,
we predict fA Z(t)dt using the conditional expectation of the integral of the diffusion
process, the optimal predictor in terms of minimizing the mean squared error, given the
observed values of the process. We obtain that, conditioning on the observed values,
the order of convergence in probability to zero of the mean squared prediction error is
Op(n- 2 ). We prove that the standardized conditional prediction error is approximately
Gaussian with mean zero and unit variance, even though the underlying diffusion is
generally non-Gaussian. Because the optimal predictor is hard to calculate exactly for
most diffusions, we present an easily computed approximation that is asymptotically
optimal. This approximation is a function of the diffusion coefficient.
I
M. Fuentes is an assistant professor at the Statistics Department, North Carolina State University, NC 27695.
Email address: [email protected].
AMS 1991 subject classifications: Primary 62M20j secondary 62M40, 41A25.
Key words and phrases. Diffusion process, fixed-domain asymptotics, infill asymptotics, numerical integration.
1. Introduction. Consider predicting
fA Z(t)dt for a diffusion process Z based on observations
Z(n) = (Z(to n ), ... , Z(t nn )), where A is a bounded interval [a, b] and a = tOn < ... < t nn = b,
n = 1,2, .... We assume the step of discretization tends to zero as n -+
00:
limn --+ oo max2~i~n (tin-
ti-l,n) = O. Let Zn(t) = E[Z(t)IZ(n)] and C~(tl,t2) = E[Z(tl)Z(t2)IZ(n)] - Zn(tl)Zn(t2) be the
conditional mean and covariance functions, respectively, for Z. For Z Gaussian, Zn(t) is a linear
function of Z(n) and c~ (tl, t2) is nonrandom. However, for diffusions, Zn(t) is generally not linear
in Z(n) and c~(tl, t2) is random.
Numerical integration over deterministic functions is studied, for example, by Davis and Rabinowitz (1984), Ghizzetti and Ossccini (1970) and Chakravarti (1970) by taking a linear function
of Z(n)' Numerical integration over random functions taking a linear function of Z(n) is done by
Marnevskaya and Jarovich (1984), Stein (1993 and 1995), Pitt, Robeva and Wang (1995), and Ritter (1995). Stein (1987) presents a non-linear approximation to the integral of a transformation of
a Brownian motion process, this is a special case of the central idea in this paper. Considering the
close relationship between predicting integrals and estimating regression coefficients from stochastic processes described by Sacks and Ylvisaker (1966), the work by Sacks and Ylvisaker (1966,
1968, 1970), Eubank, Smith and Smith (1981) and Wahba (1971, 1974) on designs for estimating
regression coefficients are also relevant for the prediction problem.
Matheron (1985) has developed a technique for approximating the unconditional distribution
of fA Z(t)dt when the process is non-Gaussian, but we are interested in making inferences about
fA Z(t)dt given Z(n), so the conditional distributions are relevant.
The conditional expectation of the integral of the diffusion process,
is the optimal predictor (in the sense of minimizing the mean squared error) of fA Z(t)dt given Z(n)'
Since this predictor is hard to calculate exactly for most diffusions, we present an approximation
1
that still yields better results than the best linear predictor (BLP) and is asymptotically optimal
as n
~
00.
The primary purpose of this paper is to show that, under certain conditions, the standardized
conditional prediction error
[var {fA Z(t)dt -
fA Zn(t)dt}] 1/2
is approximately N(O, 1), even if the underlying process is non-Gaussian. We also obtain an easily
computed approximation for the general nonlinear predictor, fA Zn(t)dt, that is a function of the
diffusion coefficient.
First, in Section 2, we show that for large n, the conditional distribution of the prediction
error is approximately Gaussian. Next, in Section 3, we show that the BLP is generally not asymptotically optimal and give an easily computed asymptotically optimal predictor. Then, in Section
4, we give an approximation for the conditional standard error of the prediction. Finally, in Section
5, we show a simulation study to assess the accuracy of this asymptotic result in finite samples,
and in Section 6 we present some final remarks.
2. Asymptotic Normality of the Prediction Error.
We predict the integral of a diffusion
over the interval A = [0,1]. It is straightforward calculation to generalize the results in this paper
to any other bounded interval in R Thus, for a homogeneous diffusion process Z on [0,1]' consider
the prediction of
1
1
Z(t)dt
based on observing Z(t) at 0= tOn
< tIn'·· < t nn = 1. Specifically, let Z(n) = (Z{tOn), ... , Z(tnn ))
and predict the integral by the optimal predictor
2
We will often write
ti
to denote
tin
for i = 0,'" ,n to simplify the notation. In this section we
will show that the standardized error in predicting the integral of a diffusion process with the
optimal predictor (in the mean squared error sense), given the observed values of the process, is
asymptotically N(O, 1).
We will make the following assumptions about the diffusion process
ZO
throughout this
paper,
(A.l)
E{Z(t + s) - Z(t)IZ(t)
= x} = sJ.t(x) + o(s)
(A.2)
E{(Z(t + s) - Z(t»2IZ(t)
(A.3)
E{IZ(t + s) - Z(t)1 3 IZ(t) = x} = s1+ 1/ 2k(x)
= x} = su 2(x) + o(s)
+ 0(s1+1/2)
where the remainder terms in (A.l)-(A.3) are uniform in x. Note that conditions A.I-A.3 require
the existence of finite conditional moments of orders 1, 2 and 3. In most practical examples of
diffusions, the indicated moments exist. It would be preferable to have assumption (A.3) in terms
of the diffusion parameters, J.t(x) and u 2(x). The diffusion parameter J.t(x) is frequently called
the drift coefficient, and u 2 (x) the diffusion coefficient. The following assumption gives explicit
hypothesis on the diffusion parameters, and implies (A.3) :
(AA)
The diffusion parameters, J.t and u 2 are bounded, (see Karatzas and Shreve, p. 367), Le.
J.t(x)
+ u2(x)
~ Pi
°~ t <
00,
and the diffusion coefficient, u 2 , is a strictly positive and uniformly bounded function
that has a continuous and uniformly bounded derivative.
We prove now that (AA) implies (A.3). For a diffusion process B(t) with constant diffusion
coefficient u 2 , we have:
(2.1)
We apply the transformation function 9 to the diffusion process Z, where g(x) = I~ u(y)-ldy.
Then 9 (Z(t» = B(t) has constant diffusion coefficient and satisfies (2.1). By (A.4) we can obtain
3
the following Taylor expansion for Z(t + s) = g-1 (B(t
+ s)) centered at t, where g-1 is the inverse
function of 9:
[Z(t + s) - Z(t)] = {g-1 (B(t))}' [B(t + s) - B(t)] +
{g-1
where
E
E
(B(t + E))}" [B(t + s) - B(t + E)]2
(D,s). Because B(t) satisfies (2.1), and the first and second derivatives of g-1 are
uniformly bounded (by the definition of 9 and (A.4)),
{g(z)-I}' = u(z)
2
{g(z)-I}" = u(z)u(z)' = {u ;z)}',
and
it follows that Z(t) satisfies (A.3).
We will sometimes require the following set of conditions for the dense triangular array of
points in the bounded interval:
F-l(~),ton =
°
(B.1)
tin =
and t nn = 1
(B.2)
F is a continuous strictly monotone cdf on [0,1] with derivative
(B.3)
f
f
is a continuous function.
A sequence that satisfies B.1 - B.3 is called a regular sequence by Sacks and Ylvisaker (1966).
We define
'lri(X,
t) to be the conditional density that from the state value x at time t the
sample path of ZO satisfies Z(tHd = Zi+l at time tHl, and make the following assumption:
(C.1)
'lri(X,
t) has two continuous derivatives with respect to x
and it is a differentiable function with respect to t.
It would be preferable to have the previous assumption, (C.1), in terms of the diffusion
coefficients. The following assumption, (C.2), has explicit hypotheses on p. and u and implies
(C.1) :
(C.2)
u(x) and p.(x) are bounded and uniformly
Holder-continuous functions
(see Karatzas and Shreve, 1991, p. 368).
4
In the following discussion
0
and 0 have the usual interpretation in terms of order of mag-
nitude statements, while op and Op mean that
0
and 0 hold respectively with a probability that
can be chosen arbitrarily close to one.
We now give the following definitions that will be needed in the proof of the main theorem
in this section:
( i) Equivalent measures. Let
with values Z(t)
Z be a diffusion process of time parameter t, where t
= Z(w, t), w
E 0, on a probability space (0,
a-algebra W is generated by Z(t) = Z(w, t) on
ET
= [0,1]
W, P). We assume that the
° as the parameter t runs through the set T.
Let PI be another measure on the a-algebra W. It is said to be absolutely continuous with
respect to P if PI (A)
= 0 whenever P(A) = 0 for A
E W. Measures PI and P are said to be
equivalent if they are mutually absolutely continuous.
( ii ) 'I'ransition density. Let p(t, x, y) denote the transition density of Z at time t. That is,
p(t, x, y)dy = p{y
< Z(t)
~ Y
+ dylZ(O) =
x}.
Then p(t, x, Zi+l) is the solution of the following backward equation,
(2.2)
where the Radon-Nikodym derivatives are supplied by the Girsanov formula (see Karatzas
and Shreve, 1991).
( iii ) Conditioned diffusion process. Let Zt be a process on [ti, ti+l] such that,
(2.3)
where £(X) denotes the distribution law of a random variable X. The conditioned diffusion
process is itself a diffusion with a time-varying drift j.£*(x, t) and same diffusion coefficient,
a 2 (x), as the original process. The transition probability density for the conditioned process
5
is given by
*(
) _ p(t - s,x,y)p(ti+l - t,y,Zi+t}
P s, t, x, Y -
(
P ti+l -
S,
X, Zi+l
).
( iv ) Prediction Error. We define PEn, the error in predicting
J; Z(t)dt given Z(n) with the
optimal predictor; that is,
Furthermore, SPEn is the standardized prediction error given Z(n),
We will prove in Theorem 2.1 that SPEn is asymptotically N(O, 1).
We present the following lemma that will be used in the proof of Theorem 2.1. We assume
that the diffusion process Z satisfies assumptions (A.1- A.3), (B1- B.3) and (C.1).
The transition density of zt for ti
pi(t, Xj s, y)dy
= p(y < Z(s)
(2.4)
=
for t
~
s
~ y
<t
~
ti+1I is given by
+ dyIZ(ti) = Zi, Z(t) = x, Z(ti+l) = Zi+ 1 )
p(s - t, x, y)1ri(y, s)dy
1ri(X, t)
< s. The drift coefficient for Zt, for ti < t < ti+ 11 is
oo
/-Li(x, t) = lim -hI
h..j.O
r+ (y J-oo
x)pi(t, Xj t
+ h, y)dy.
LEMMA 2.1
The conditioned diffusion process zt has a non-homogeneous infinitesimal mean:
(2.5).
6
The diffusion coefficient of the process Z; is
(}"2,
the diffusion coefficient of the process Z.
Proof of Lemma 2.1:
Lemma 2.1 is a known result (see Karlin and Taylor, 1981, p. 267-268). The proof is very
straightforward once we have the following Taylor expansion for 7ri(X, t):
(2.6)
7ri(Y, t
+ h) = 7ri(X, t) + (y -
a7r"
x) a: (x, t)
+h
a7r"
at' (x, t)
+ o(y -
x)
+ o(h).
The Main Theorem
THEOREM 2.1
Consider predicting the integral of a diffusion process Z over [0,1], based on the observations
1
~
Z(n) = (Z(tl n ), . .. ,Z(tnn )), by Jo Zn(t)dt. Suppose
(i)
the parameters of the diffusion process Z satisfy relations (A.1 - A.3)
(ii)
conditions (B.1- B.3) are satisfied by the sequence of points in [0,1]
(iii)
the conditional probabilities 7ri satisfy condition (C.1).
Then conditional on Z(n)
SPEn ~ N(O, 1)
(2.7)
with probability 1.
Proof of Theorem 2.1:
Given Z(n), PEn is a sum of independent, mean zero random variables, so this is a triangular
array situation. We study the conditional cumulants of orders 1 to 3 of the prediction error
7
because, applying Lyapounov's condition (Billingsley, 1995) for 0
=
1, we can prove the weak
convergence of the distributions. Suppose Yin"", Ynn are independent random variables, such
Therefore the problem is reduced to studying just the cumulants of ft~~l Zi(t)dt given Zi at time
ti, where the process
ZiO
is defined in (2.3). The infinitesimal mean of Zi satisfies
(2.9)
Recall that we obtained an expression for
J.ti
in Lemma 2.1.
In Lemma 2.1 we also showed that the variance of Zi is
(2.10)
In addition to the infinitesimal relations (2.8) and (2.9), the following higher-order infinitesimal
moment relation is satisfied when (A.3) holds:
(2.11)
All this means that
(2.12)
n-l
= ~)ti+l
- ti)3{u 2 (Zi)j12}
+ o(n- 2 ).
i=Q
We get the first equality by applying (2.8), and the last expression by integrating the infinitesimal variance of Zi(t) when t belongs to [ti, ti+1], where by assumptions (A.1) - (A.3), the
remainder term in (2.12) is uniform in
Z(n)'
Therefore, the order of the conditional standard error
8
of the prediction error is O(n- 1 ) as n --+
00.
It follows from Eq. (2.12) and conditions (B.1- B.3)
that n[var{PEnIZ(n)}] 1/2 converges in probability to the random variable L
as n
t
00,
= fo1 a2(Z(t))w(t)dt
where w(t) = 112f(t)-3.
Using the same argument as the one leading to Eq. (2.12) together with (2.11), we get that
Let
s~ (Z(n»)
be the conditional variance of Sn (Z(n») = (Xln+' .+Xnn)-E(X 1n +·· ,+XnnIZ(n»)
given Z(n)' By Eqs. (2.11) and (2.12), we get
Sn
z(n)
Bn
z(n)
which is Lyapounov's condition for 8 = 1. Since Lyapounov's condition holds,
L
---+
N(O, 1). If we write the result in terms of the diffusion process Z, then relation (2.7) holds and
the conditional prediction error is asymptotically normal.
In the following corollary whose proof is obvious, we reformulate the assumptions of Theorem 2.1
in terms of the diffusion parameters, J.t(x) and a 2 (x).
COROLLARY 2.1.
Theorem 2.1 still holds if (A.S) is replaced by (A.4) in (i) and (C.l) is replace by (C.2) in
(iii).
9
3.
Approximating the Optimal Predictor. The conditional expectation of the integral of
the diffusion process is the optimal predictor of
Jo1 Z(t)dt given Zen).
But because this predictor
is hard to calculate exactly for most diffusions, we present the following approximation to it that
yields an asymptotically optimal predictor:
1
n
(3.1)
In(Z(n») = 2::(tin - ti-1,n)Zi
n
+ 2n 2::(tin -
i=l
2
ti_1,n){a (Zi)}'.
i=l
Here, we write Zi to denote Z(tin) to simplify the notation, and {a 2 (Z)}' to denote the derivative
of a 2 with respect to Z. Approximation (3.1) is a linear function of Z(n) , ~~=1 (tin - ti-1,n)Zi, plus
a nonlinear term in Zen) that is a function of the derivative of the diffusion coefficient, and could
be thought of as the adjustment for the conditional bias of the BLP.
We define PEn' the error in predicting
J; Z(t)dt with In(Z(n»),
In this section we will show that the error PEn - PEn, in approximating
Jo1 Zn(t)dt with
In(Z(n»), is negligible compared to the conditional standard deviation of the prediction error PEn.
Then, by Theorem 2.1, the prediction error PEn is asymptotically normal.
THEOREM 3.1
Under assumptions (A.t)-(A.3), (B.t)-(B.3), and (C.t), conditional on Zen)
with probability t.
We now present a proposition that will be used in the proof of Theorem 3.1.
10
PROPOSITION 3.1
Under conditions (A.l-A.3), (B.l-B.3) and (C.l),
Proof of proposition 3.1:
We now prove that the remainder term when we approximate
with In(Z(n» is of order L:~==-OI{ F- 1 (~) _F- 1 (*)} 2+6 for some c5
> O. By conditions (B.1- B.3),
we get that
Thus, the error in the approximation, PEn - PEn, is negligible compared to the conditional
standard deviation of PEn' which we will prove in proposition 4.1 is Op
(n- 1 )
•
By the Markov property of the diffusion process Z,
Letting Z = (zo,···, zn) be the observed values of ZO at times to,···, tn, by straightforward
calculation
1
(3.2)
1
o
E(Z(t)IZ(n) = z)dt =
n-l1tHl
L E(Z(t) i=O
t,
n-l
+L
Zi(ti+1 - ti).
i=O
11
Z(ti)IZ(ti) = Zi, Z(ti+d = Zi+1)dt
In (2.4) we showed that
J.Li, the transition density for the conditioned process, was a function
of 1ri (that satisfies (C. 1)) and p, the transition density of Z. So, we need an explicit expression for p.
We transform the coordinates applying the transformation function g, where g(x) =
I:a u(y)-ldy.
Then g(Z) is a process with constant diffusion coefficient.
Now, if P is the probability measure of the diffusion process Z, let P be a probability measure
such that P and P are equivalent, and under P, Z has drift coefficient
(3.3)
It is routine to verify that the process B = g(Z) is a Brownian motion under P. Therefore, by a
change of variables we get an equation equivalent to (2.2),
8p(t, b, bi+l)
1 8 2 p(t, b, bi+1)
8t
- 2
8b 2
(3.4)
The solution is a Gaussian density. By a change of variables again, we can express the solution in
terms of the process Z (under P) and obtain
and
Assumption (C.1) postulates sufficient regularity for 1ri(X, t) to permit the use of the following
Taylor expansion:
(3.5)
where
f
E
(x, y).
12
We define Ri the residual term when we approximate
ft:
H1
Z(t)dt with the conditionally
optimal predictor restricted to the interval [ti+l, til E [0,1]. By the definition of p*(t, Xj 5, y) in
(2.4) and the Taylor expansion (3.5), we obtain
By a change of variables again, we can get an explicit expression for the second derivative of
1ri(X, t) with respect of of x, using the same argument as in (3.4). By assumption A.3, Ri is of
order Op (n-(2+O)) , where 8 is the same as in (A.3). Thus, we obtain
Then we get that (under P)
1
1
n-l
Zn(t)dt = L
o
n-l
Zi(ti+l - ti)
+ L(ti+l -
i=l
-I: al(Zi~a(Zi)
(3.6)
ti)(Zi+l - Zi)
i=l
(ti+l - ti)2
+ Op (n-(l+6))
i=l
Thus, under the probability measure P,
n-l
n-l
n(1+o/2) Z(n) (t)dt - ~ Zi(ti+l - ti)
(
(3.7)
_
+ ~(ti+l -
ti)(Zi+l - Zi)
~ 07' (Z';o7(Z,) (ti+l _ 1;)2) ~ n(l+'!2) (PEn - PEn)
p. O.
It is straightforward consequence of the equivalence of P and P that (3.7) holds under P as
well. In other words,
n(l+o/2) (PEn - PEn) ~ O.
13
Thus, we approximate the optimal predictor of the integral
Jol Zn(t)dt with In(Z(n)) and the
error in this approximation, PEn - PEn is negligible compared to the standard deviation of PEn-
Proof of Theorem 3.1:
We proved in Theorem 2.1 that the standardized prediction error SPEn is, conditional on
Z(n) , asymptotically N(O, 1). By Propositions 3.1 and 4.1, PEn - PEn is negligible compared to
the conditional standard deviation of PEn, so Theorem 3.1 follows.
A symmetric approximation for the optimal predictor.
We present a symmetric approximation for the predictor that is invariant if the order of time
for the diffusion on the interval [0,1] is reversed. We get this approximation as the average of two
approximations for the optimal predictor. The first approximation is obtained by assuming that
we observe the diffusion process Z on [0,1] starting at time 0, that is, to =
°
second approximation, we assume the observations start at time 1, that is, to
and t n = 1. For the
= 1, and t~ = 0.
obtain
n-I
n-I
In (Z(n)) = L Zi(ti+1 - ti-I)
i=2
(3.8)
1
+ 2 [ZI(t2 -
tt)
+ L(ti+1 -
ti-I)(Zi+1 - Zi-I)
i=2
+ Zn(tn -
tn-I)]
1
+ 2 [(t2 - tl)(Z2 - ZI) + (t n - tn-I)(Zn - Zn-I)]
_ u'(ZI)U(ZI) (t2 _ tI)2 _ U'(Zn)U(Zn) (tn - t _I)2.
n
4
4
Thus,
14
We
4. The Variance of the Prediction Error.
In the following proposition we prove that the
order of convergence to zero of the conditional variance of the prediction error is Op(n- 2 ). Then
the order of convergence to zero of the error in approximating
J; Zn(t)dt by In(Z(n)) is faster than
the order of the conditional standard deviation of PEn.
PROPOSITION 4.1
As n
~ 00,
(4.1)
Proof of proposition ·401:
By the Markov property of the diffusion process and using the same argument as in (3.2), the
variance of the prediction error, when conditions (A.1- A.3), (B.1 - B.3) and (C.1) are satisfied,
can be written as
(4.2)
n-l
= Z)ti+l -
ti)3{u 2 (Z(ti))j12}
+ op(n- 2 ).
i=O
Where the process
zt is defined in (2.3).
Then the order of the standard error of the prediction
error is Op(n- 1 ).
15
A symmetric approximation for the variance of the prediction error.
Eq. (4.2) gives us an approximation for the conditional variance which can also be expressed
in a symmetric manner
(4.3)
5. Simulation Results with Known Diffusion Parameters.
To assess the accuracy of
the proposed approximations to the integral of a diffusion process Z and to the variance of the
prediction error in finite samples, we performed several simulation experiments, assuming that the
diffusion coefficient was a known function.
We simulated n values of a diffusion process Z at times to, . .. ,tn-l where ti - n~l' for
i = 0, 1, ... , n - 1, n = 100 and Z is a transformation of a Brownian motion B,
Z t _
exp{B(t)}
( ) - [1 + exp{B(t)}]
(5.1)
This diffusion process is a uniformly bounded process and, hence, trivially satisfies the assumptions
(AA) and (C.2).
The parameters of the diffusion Z, as defined in (5.1), can be written as a function of the
Brownian motion process. The infinitesimal variance of Z is
(T2(Z(t)) =
exp{2B(t)} .
[1 + exp{B(t)}]4
Neither In(Z(n»), nor the approximation for the standard error, in the prediction error depend
on the drift coefficient of Z, which can be an unknown function. So, for the asymptotics, we do not
16
need an expression for the infinitesimal mean of Z. Clearly, In(Z(n») is a function of the infinitesimal
variance of Z and also of the derivative of that parameter, given by
2 [1 - exp{B(t)}] exp{B(t)}
[1 + exp{B(t)}j3
We conducted the following simulation experiment to assess the asymptotic normality of the
conditional prediction error:
( i ) We simulated n values of a Brownian motion and applied the transformation (5.1) to
get n observations of the diffusion process Z. We show in Figure 5.1 Zen), the n simulated
values of Z, for a single realization of Z. Using Eq. (3.8) and (2.12) we get the value of the
predictor and the conditional variance of the prediction error, because the diffusion coefficient
is known.
( ii ) In order to simulate the distribution of the prediction error, we need the value of the
predictor obtained in (i), the conditional variance of the prediction error obtained also in (i),
1
1
and fo Z(t)dt, which is unknown. Then we have to simulate fo Z(t)dt conditional on Zen)'
In order to obtain values of f~ Z(t)dt, we simulated m values of Z(t) at m equally spaced
locations in time in every interval [ti, ti+1], conditioning on the observed values Zen)' We
conducted several simulation experiments with different values for m, and m = 10 seemed
to be sufficiently large in the sense that increasing m has only a negligible effect on the
results. Figure 5.2 shows a single realization of the IOn simulated values of Z conditional
on the n observed values of Zen)' We considered the mean of the IOn simulated values of Z
conditioning on Zen) to be the unknown quantity that we want to predict, f~ Z(t)dt.
1
( iii) We repeated step (ii) k times to obtain k simulated values for fo Z(t)dt. We also obtained
k simulated values of the prediction error
PEn and the variance of PEn, always conditioning
on the n observed values Zen)'
17
( iv ) We plotted the histogram of the k simulated values of the prediction error
Figure 5.3 (a)). Here k
PEn
(see
= 100.
Then we repeated steps (i)-(iv) 100 times to study the conditionally asymptotic normality,
but conditioning now on other simulated values of Zen).
18
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+
o
LO
c:i
+
+
c:i
+
+
+
+
"(j)
::::J
==
is
+
++
+
+
+
+
+
+
+
I
I
I
I
I
I
0.0
0.2
0.4
0.6
0.8
1.0
Time
Figure 5.1. Simulation of Z(n), the observed values ofthe diffusion process Z, with n = 100.
19
I/)
co -
o
o
CO
o
-
t
S
N
I/)
I/)
o
-
»s
'0
o
.. 0
0
~oo
It'b~
o
0
o
I/)
o
o
0
8
000
o
-
:
o
:t;
I/)
o..q-~-r-------.--------.-----...,------.,...-----rI
I
I
I I
I
0.0
0.2
0.4
0.6
0.8
1.0
Time
1
Figure 5.2.
Simulation of J0 Z(t)dt conditional on Zn. The vertical axis shows 1000
simulated values of Z conditional on the n=100 observed values Z(n)' The average of the 1000
simulated values approximates J~ Z(t)dt. The symbol
20
+ is used for the values of Z(n)'
Figure 5.3 (a) is a histogram of the distribution of the prediction error using the proposed
approximation for the optimal predictor,
In(Z(n)),
for the particular
Z(n)
= Z(n)
showed in Figure
5.1. Because the histogram is centered at zero and has a normal shape, this provides some evidence
that the proposed predictor is conditionally unbiased and asymptotically normal. In this simulation
study n
= 100, m = 10 and k = 100.
In the normality plot, Figure 5.4, we can see evidence of this
asymptotic normality.
Figure 5.3 (b) shows the distribution of the prediction error when the predictor is the classic
linear combination of the observed values. The vertical line in the graph is located at the mean value
of the sampling distribution. The histogram is not centered at zero, which reflects the fact that
this naive predictor, a linear combination of the observed values, has nontrivial conditional bias in
this case (the value of the conditional bias is .00087), which is approximately 2~
Jo1 a(Zt)a'(Zt)dt.
Now we compare the mse of the linear and nonlinear predictor, for the particular
Z(n) = z(n)
pictured in Figure 5.1, to show that the nonlinear predictor is better. The mse of the nonlinear
predictor is 4.13 x 10-7 and the mse of the linear predictor is 8.01 x 10-7, almost twice as large.
In order to show that this result is not just due to a favorable value of
the mse for more simulations conditioning on other values of
nonlinear predictor from the simulation conditional on the
Z(n)'
Z(n)
Z(n) ,
Table 5.1 compares
We compared the mse of the
shown in Figure 5.1 with the one
obtained using the asymptotic approximation presented in Theorem 3.1, and the asymptotic mse
is 3.97
X
10-7, very close to the mse of the nonlinear predictor and roughly half the mse of the
linear predictor.
21
Prediction Error for Nonlinear Predictor
LO
..-
~
c::
Q)
:::l
0
..-
C"
~
U.
LO
o
-0.0015
-0.0010
-0.0005
0.0
0.0005
0.0010
-
0.0015
Figure a
Prediction Error for BLP
0
C\I
LO
..-
>0
c::
Q)
:::l
0"
0
..-
~
U.
LO
o
-0.0005
0.0
0.0005
0.0010
0.0015
-
0.0020
Figure b
Figure 5.3. Distribution of the approximated prediction error, u 2 known. (a) Distribution of
1
the conditional prediction error when predicting
Z{t)dt given Z(n) = z(n) with the optimal
predictor. The vertical axis represents the frequencies out of 100 simulations. (b) Distribution
of the prediction error given Z(n) = z(n) using the BLP. The conditional bias is .00087.
Jo
22
•••• n
• • _ . __
.
•••••••••••
.
__
.• . .
__ •
•••••••••••••••••
:
:
...e
...
w
c
o
~
~
o
mmmmmmm1mmm
0""0
~
'EI'd
""0
C
I'd
en
.,...
I
•
••
-2
•
o
-1
1
2
Quantiles of Standard Normal
Figure 5.4. Normality plot for the approximated prediction error,
for the standardized conditional prediction error when we predict
with the approximately optimal predictor, In(Z(n»)'
Io
23
known. Normality plot
Z(t)dt given Zen) = Zn
(72
1
We also obtained naive 95% prediction intervals. For the nonlinear predictor in the simulation
shown in Figure 5.3, conditioning on the particular value of Z(n)
= Z(n)
pictured in Figure 5.1,
the 95% prediction interval, using the approximation for the conditional variance (4.3) presented
in Section 4, is (0.572687, 0.575567). We calculated the 95% prediction interval for the linear
predictor acting as if we have a Brownian motion, and estimated
0: 2 =
(5.2)
(72
with
n
L (Z(ti) -
Z(ti_l))2 .
i=l
This estimator is uniformly minimum variance unbiased for
(72
under the assumption that Z is a
Brownian motion. Thus, we could use n -20: 2 to build the prediction interval. For this simulation,
using the value of Z(n) = Z(n) plotted in Figure 5.1, the value of n- 20: 2 is 4.52 x 10- 7 . The
prediction interval for the linear predictor using n- 2 0: 2 is (0.573507, 0.576137). Thus, the interval
for the linear predictor is approximately the interval for the nonlinear predictor shifted to the right
by around .0006, or by approximately the conditional bias for the linear predictor.
We repeatedly simulated
Jo1 Z(t)dt, conditioning on the same Z(n) as in Figure 5.1, to see how
well the obtained prediction intervals work. For this simulation we used the technique presented
in part (ii) of this section. In 450 out of 500 simulations (90% of the time), the prediction interval
for the nonlinear predictor contains
Jo1 Z(t)dtj
whereas, in 390 out 500 simulations (78% of the
time), the prediction interval for the linear predictor contains
the simulated
Jo1 Z(t)dt.
The median value of
Jo1 Z(t)dt is 0.574027, approximately the mid point of the interval for the nonlinear
predictor. All these results suggest that the prediction interval, obtained when we use the nonlinear
predictor and the approximate conditional standard prediction error, has somewhat better coverage
than the one obtained for the linear predictor (see Figure 5.5).
24
Conditional Distribution
o
C\I
~
...... nonlinear predictor
--
linear predictor
o
lX)
o
C\I
;
: I
o
I
I
I
I
I
0.572
0.573
0.574
0.575
0.576
Integral of the Diffusion
Figure 5.5.
Prediction intervals for I~ Z(t)dt.
I~ Z(t)dt given a fixed Z(n)
=
Simulated distribution of the integral
Z(n). The solid vertical line is located at the sample mean
of the simulated values of I~ Z(t)dt. The marks on the horizontal axis are the prediction intervals for the linear and nonlinear predictors. The vertical axis shows the frequencies out of
500 simulations.
25
Because the results obtained could be just due to a favorable realization of
Z(n),
we ran
many other simulations. We present the mse of the linear predictor obtained by using two different
approaches (using the conditional variance for the linear predictor and using n- 2 o: 2 ) and the mse
of the nonlinear predictor using (4.3), for 10 simulated
Z(n),
with n = 500, assuming the diffusion
coefficient is a known function.
Conditional mse for the linear and nonlinear predictors:
Linear
Nonlinear
1.18
1.04
1.09
0.31
1.86
0.61
2.07
0.52
1.70
1.09
2.23
1.42
2.02
0.04
1.99
0.61
1.98
0.29
1.92
0.79
Table 5.1 Conditional mse for the linear
and nonlinear predictors multiplied by 108 .
26
mse for the best linear predictor using the conditional
variance using n- 2 o: 2 :
Conditional mse
mse using n- 2 o: 2
1.18
1.14
2.09
2.10
1.86
2.02
2.07
2.14
1.70
1.64
2.23
2.26
2.02
1.97
1.99
2.02
1.98
1.98
1.92
2.03
Table 5.2 mse values multiplied
by 108 for the linear predictor.
Table 5.1 shows in the first column the mse for the linear predictor using the conditional
variance for the linear predictor. The second column is the mse for the nonlinear predictor using
the approximation to the conditional variance presented in Section 4 (4.3), assuming
(72
is known.
The first column have mse values that are larger than the mse values for the nonlinear predictor.
Table 5.2 shows in the first column the mse, conditioning on the same values for
Z(n)
as in
Table 5.1, for the linear predictor using the conditional variance for the linear predictor, and the
second column shows the mse for the linear predictor acting as if it is a Brownian motion and using
n- 2 o: 2 • The two columns have similar mse values, which indicates that using n- 2 o: 2 to compute
the mse for the linear predictor is an appropriate approach.
27
6.
Final Remarks.
In this paper we have presented the asymptotic normality of the
prediction error for predicting the integral of a diffusion process with the integral of the conditional
expectation of the process, the optimal predictor. We have also obtained a simple asymptotically
optimal approximation for the predictor and a simple asymptotically valid approximation for the
variance of the prediction error, but in both cases we assumed that the diffusion coefficient was a
known function.
1
We could generalize Theorem 2.1 to other additive functionals, for instance fo g(Zt)dt, because, by the Ito transformation formula we easily obtain the diffusion parameters of yt = g(Zt) as
a function of Zt through g. If 9 is smooth enough so that assumptions (AA) and (C.2) are satisfied
by the new diffusion parameters, then Theorem 2.1 holds for the diffusion process g(Zt), and we
1
obtain the asymptotic normality of fo {g(Zt) - E [g(Zt)lg (Z(n))]} dt.
Acknowledgments. The author is grateful to Michael L. Stein for his many valuable comments,
insights and guidance in obtaining the results in this paper.
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